Properties of Modified Riemannian Extensions

Let M be an n-dimensional differentiable manifold with a symmetric connection ∇ and T*M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension ğ∇, c on T*M defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T*M...

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Дата:	2015
Автори:	Gezer, A., Bilen, L., Cakmak, A.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2015
Назва видання:	Журнал математической физики, анализа, геометрии
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Цитувати:	Properties of Modified Riemannian Extensions / A. Gezer, L. Bilen, A. Cakmak // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 159-173. — Бібліогр.: 24 назв. — англ.

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spelling	irk-123456789-1180242017-05-29T03:02:22Z Properties of Modified Riemannian Extensions Gezer, A. Bilen, L. Cakmak, A. Let M be an n-dimensional differentiable manifold with a symmetric connection ∇ and TM be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension ğ∇, c on TM defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which TM endowed with the horizontal lift HJ of an almost complex structure J and with the metric ğ∇, c is a Kähler-Norden manifold. Also curvature properties of the Levi-Civita connection of the metric ğ∇, c are presented. Пусть M - n-мерное дифференцируемое многообразие с симметричной связностью ∇ а TM - его кокасательное расслоение. В статье изучены некоторые свойства модифицированного риманова расширения ğ ∇, c на TM, которое определяется с помощью симметричного (0, 2)-тензорного поля c на M. Получены условия, при которых TM, наделенное горизонтальным лифтом HJ почти комплексной структуры J и метрикой ğ∇, c, является многообразием Kэлера-Нордена. Также представлены свойства кривизны связности Леви- Чивита метрики ğ ∇ , c. Нехай M — n-мірний диференційовний многовид із симетричного зв'язністго ∇, а TM — його кодотичне розшарування. В статті вивчено деякі властивості модифікованого ріманова розширення ğ ∇, c на TM, яке визначається за допомогою симетричного (0, 2)-тензорного поля с на М. Отримано умови, за яких T*M, наділене горизонтальним ліфтом HJ майже комплексної структури J і метрикого ğ∇, c, є многовидом Келера-Нордена. Також представлені властивості кривини зв'язності Леві-Чівіти метрики ğ∇, c. 2015 Article Properties of Modified Riemannian Extensions / A. Gezer, L. Bilen, A. Cakmak // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 159-173. — Бібліогр.: 24 назв. — англ. 1812-9471 DOI: 10.15407/mag11.02.159 MSC2000: 53C07, 53C55, 53C35 http://dspace.nbuv.gov.ua/handle/123456789/118024 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let M be an n-dimensional differentiable manifold with a symmetric connection ∇ and TM be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension ğ∇, c on TM defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T*M endowed with the horizontal lift HJ of an almost complex structure J and with the metric ğ∇, c is a Kähler-Norden manifold. Also curvature properties of the Levi-Civita connection of the metric ğ∇, c are presented.
format	Article
author	Gezer, A. Bilen, L. Cakmak, A.
spellingShingle	Gezer, A. Bilen, L. Cakmak, A. Properties of Modified Riemannian Extensions Журнал математической физики, анализа, геометрии
author_facet	Gezer, A. Bilen, L. Cakmak, A.
author_sort	Gezer, A.
title	Properties of Modified Riemannian Extensions
title_short	Properties of Modified Riemannian Extensions
title_full	Properties of Modified Riemannian Extensions
title_fullStr	Properties of Modified Riemannian Extensions
title_full_unstemmed	Properties of Modified Riemannian Extensions
title_sort	properties of modified riemannian extensions
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/118024
citation_txt	Properties of Modified Riemannian Extensions / A. Gezer, L. Bilen, A. Cakmak // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 159-173. — Бібліогр.: 24 назв. — англ.
series	Журнал математической физики, анализа, геометрии
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2015, vol. 11, No. 2, pp. 159–173 Properties of Modified Riemannian Extensions A. Gezer1, L. Bilen2, and A. Cakmak1 1Ataturk University, Faculty of Science, Department of Mathematics 25240, Erzurum-Turkey E-mail: agezer@atauni.edu.tr ali.cakmak@atauni.edu.tr 2Igdir University, Igdir Vocational School 76000, Igdir -Turkey E-mail: lokman.bilen@igdir.edu.tr Received January 21, 2014, revised December 16, 2014 Let M be an n-dimensional differentiable manifold with a symmetric connection ∇ and T ∗M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension g̃∇,c on T ∗M defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T ∗M endowed with the horizontal lift HJ of an almost complex structure J and with the metric g̃∇,c is a Kähler–Norden manifold. Also curvature properties of the Levi–Civita connection of the metric g̃∇,c are presented. Key words: cotangent bundle, Kähler–Norden manifold, modified Rie- mannian extension, Riemannian curvature tensors, semi-symmetric mani- fold. Mathematics Subject Classification 2010: 53C07, 53C55, 53C35. 1. Introduction Let M be an n-dimensional differentiable manifold and T ∗M be its cotan- gent bundle. There is a well-known natural construction which yields, for any affine connection ∇ on M , a pseudo-Riemannian metric g̃∇ on T ∗M. The met- ric g̃∇ is called the Riemannian extension of ∇. Riemannian extensions were originally defined by Patterson and Walker [15] and further studied by Afifi [2], thus relating pseudo-Riemannian properties of T ∗M with the affine structure of the base manifold (M,∇). Moreover, Riemannian extensions were also con- sidered by Garcia-Rio et al. in [8] in relation to Osserman manifolds (see also Derdzinski [5]). Since Riemannian extensions provide a link between affine and pseudo-Riemannian geometries, some properties of the affine connection∇ can be c© A. Gezer, L. Bilen, and A. Cakmak, 2015 A. Gezer, L. Bilen, and A. Cakmak investigated by means of the corresponding properties of the Riemannian exten- sion g̃∇. For instance, ∇ is projectively flat if and only if g̃∇ is locally conformally flat [2]. For Riemannian extensions, also see [1, 7, 9, 11, 12, 17, 19, 21, 22]. In [3, 4], the authors introduced a modification of the usual Riemannian extensions which is called the modified Riemannian extension. Let M2k be a 2k-dimensional differentiable manifold endowed with an almost complex structure J and a pseudo-Riemannian metric g of signature (k, k) such that g(JX, Y ) = g(X,JY ) for arbitrary vector fields X and Y on M2k. Then the metric g is called the Norden metric. Norden metrics are referred to as anti-Hermitian metrics or B-metrics. The study of such manifolds is interesting because there exists a difference between the geometry of a 2k-dimensional almost complex manifold with Hermitian metric and the geometry of a 2k-dimensional almost complex manifold with Norden metric. A notable difference between Nor- den metrics and Hermitian metrics is that G(X, Y ) = g(X,JY ) is another Norden metric, rather than a differential 2-form. Some authors considered almost com- plex Norden structures on the cotangent bundle [6, 13, 14]. In this paper, we will use a deformation of the Riemannian extension on the cotangent bundle T ∗M over (M,∇) by means of a symmetric tensor field c on M , where ∇ is a symmetric affine connection on M . The metric is the so- called modified Riemannian extenson. In Section 3, in the particular case where ∇ is the Levi–Civita connection on a Riemannian manifold (M, g), we get the conditions under which the triple (T ∗M,HJ, g̃∇,c) is a Kähler–Norden manifold, where HJ is the horizontal lift of an almost complex structure J and g̃∇,c is the modified Riemannian extension. Section 4 deals with curvature properties of the Levi–Civita connection of the modified Riemannian extension g̃∇,c. Throughout this paper, all manifolds, tensor fields and connections are always assumed to be differentiable of class C∞. Also, we denote by =p q(M) the set of all tensor fields of type (p, q) on M , and by =p q(T ∗M) the corresponding set on the cotangent bundle T ∗M . The Einstein summation convention is used, the range of the indices i, j, s being always {1, 2, . . . , n}. 2. Preliminaries 2.1. The cotangent bundle Let M be an n-dimensional smooth manifold and denote by π : T ∗M → M its cotangent bundle whose fibres are cotangent spaces to M. Then T ∗M is a 2n-dimensional smooth manifold and some local charts induced naturally from local charts on M can be used. Namely, a system of local coordinates ( U, xi ) , i = 1, . . . , n in M induces on T ∗M a system of local coordinates ( π−1 (U) , xi, xi = pi ) , i = n + i = n + 1, . . . , 2n, where xi = pi are the components of covectors p in 160 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions each cotangent space T ∗xM, x ∈ U with respect to the natural coframe { dxi } . Let X = Xi ∂ ∂xi and ω = ωidxi be the local expressions in U of a vector field X and a covector (1-form) field ω on M , respectively. Then the vertical lift V ω of ω, the horizontal lift HX and the complete lift CX of X are given, with respect to the induced coordinates, by V ω = ωi∂i, (2.1) HX = Xi∂i + phΓh ijX j∂i (2.2) and CX = Xi∂i − ph∂iX h∂i, where ∂i = ∂ ∂xi , ∂i = ∂ ∂xi and Γh ij are the coefficients of a symmetric (torsion-free) affine connection ∇ in M. The Lie bracket operation of vertical and horizontal vector fields on T ∗M is given by the formulas    [ HX,H Y ] = H [X, Y ] +V (p ◦R(X, Y ))[ HX,V ω ] = V (∇Xω)[ V θ,V ω ] = 0 (2.3) for any X, Y ∈ =1 0(M) and θ, ω ∈ =0 1(M), where R is the curvature tensor of the symmetric connection ∇ defined by R (X,Y ) = [∇X ,∇Y ] − ∇[X,Y ] (for details, see [24]). 2.2. Expressions in the adapted frame We insert the adapted frame which allows the tensor calculus to be efficiently done in T ∗M. With the symmetric affine connection ∇ in M , we can introduce the adapted frames on each induced coordinate neighborhood π−1(U) of T ∗M . In each local chart U ⊂ M , we write X(j) = ∂ ∂xj , θ(j) = dxj , j = 1, . . . , n. Then from (2.1) and (2.2), we can see that these vector fields have, respectively, the local expressions HX(j) = ∂j + paΓa hj∂h , V θ(j) = ∂ j with respect to the natural frame { ∂j , ∂ j } . These 2n-vector fields are linearly independent and they generate the horizontal distribution of ∇ and the vertical Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 161 A. Gezer, L. Bilen, and A. Cakmak distribution of T ∗M , respectively. The set { HX(j), V θ(j) } is called the frame adapted to the connection ∇ in π−1(U) ⊂ T ∗M . By putting Ej = HX(j), (2.4) Ej = V θ(j), we can write the adapted frame as {Eα} = { Ej , Ej } . The indices α, β, γ, . . . = 1, . . . , 2n indicate the indices with respect to the adapted frame. Using (2.1), (2.2) and (2.4), we have V ω = ( 0 ωj ) (2.5) and HX = ( Xj 0 ) (2.6) with respect to the adapted frame {Eα} (for details, see [24]). By the straight- forward calculations, we have the lemma below. Lemma 1. The Lie brackets of the adapted frame of T ∗M satisfy the follow- ing identities: [Ei, Ej ] = psR s ijl El,[ Ei, Ej ] = −Γj ilEl,[ Ei, Ej ] = 0, where R s ijl denote the components of the curvature tensor of the symmetric con- nection ∇ on M . 3. Kähler–Norden Structures on the Cotangent Bundle We first give the definition of pure tensor fields with respect to a (1, 1)-tensor field J . Definition 1. For a (1, 1)-tensor field J , the (0, s)-tensor field t is called pure with respect to J if t(JX1, X2, . . . , Xs) = t(X1, JX2, . . . , Xs) = . . . = t(X1, X2, . . . , JXs) for any X1, X2, . . . , Xs ∈ =1 0(M). For more information about the pure tensor, see [16, 20, 23]. 162 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions An almost complex Norden manifold (M, J, g) is a real 2k-dimensional differ- entiable manifold M with an almost complex structure J and a pseudo-Riemannian metric g of neutral signature (k, k) such that g(JX, Y ) = g(X, JY ) for all X,Y ∈ =1 0(M), i.e., g is pure with respect to J . A Kähler–Norden (anti- Kähler) manifold can be defined as a triple (M, J, g) which consists of a smooth manifold M endowed with an almost complex structure J and a Norden metric g such that ∇J = 0, where ∇ is the Levi–Civita connection of g. It is well known that the condition ∇J = 0 is equivalent to the C-holomorphicity (analyticity) of the Norden metric g [10], i.e., ΦJg = 0, where ΦJ is the Tachibana operator [16, 20, 23]: (ΦJg)(X,Y, Z) = (JX)(g(Y, Z)) − X(g(JY, Z)) + g((LY J)X, Z) + g(Y, (LZ J)X) for all X, Y, Z ∈ =1 0(M). Also note that G(Y, Z) = g(JY, Z) is the twin Norden metric. Since in dimension 2 a Kähler–Norden manifold is flat, we assume in the sequel that n = dim M ≥ 4. Next, for a given symmetric connection ∇ on an n-dimensional manifold M , the cotangent bundle T ∗M can be equipped with a pseudo-Riemannian metric g̃∇ of signature (n, n): the Riemannian extension of ∇ [15], given by g̃∇(CX,C Y ) = −γ(∇XY +∇Y X), where CX,C Y denote the complete lifts to T ∗M of vector fields X, Y on M . Moreover, for any Z ∈ =1 0(M), Z = Zi∂i, γZ is the function on T ∗M defined by γZ = piZ i. The Riemannian extension is expressed by g̃∇ = ( −2phΓh ij δi j δj i 0 ) with respect to the natural frame. Now we give a deformation of the Riemannian extension above by means of a symmetric (0, 2)-tensor field c on M whose metric is called the modified Riemannian extension. The modified Riemannian extension is expressed by g̃∇,c = g∇ + π∗c = ( −2phΓh ij + cij δi j δj i 0 ) (3.1) with respect to the natural frame. It follows that the signature of g̃∇,c is (n, n). Denote by ∇ the Levi–Civita connection of a semi-Riemannian metric g. In this section, we will consider T ∗M equipped with the modified Riemannian exten- sion g̃∇,c over a pseudo-Riemannian manifold (M, g). Since the vector fields HX and V ω span the module of vector fields on T ∗M , any tensor field is determined Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 163 A. Gezer, L. Bilen, and A. Cakmak on T ∗M by their actions on HX and V ω. The modified Riemannian extension g̃∇,c has the following properties: g̃∇,c(HX,H Y ) = c(X, Y ), (3.2) g̃∇,c(HX,V ω) = g∇,c(V ω,H X) = ω(X), g̃∇,c(V ω,V θ) = 0 for all X, Y ∈ =1 0(M) and ω, θ ∈ =0 1(M), which characterize g̃∇,c. The horizontal lift of a (1, 1)-tensor field J to T ∗M is defined by HJ(HX) = H(JX), (3.3) HJ(V ω) = V (ω ◦ J) for any X ∈ =1 0 (M) and ω ∈ =0 1 (M). Moreover, it is well known that ıf J is an almost complex structure on (M, g), then its horizontal lift HJ is an almost complex structure on T ∗M [24]. Now we prove the following theorem. Theorem 1. Let (M, J, g) be a Kähler–Norden manifold. Then T ∗M is a Kähler-Norden manifold equipped with the modified Riemannian extension g̃∇,c and the almost complex structure HJ if and only if the symmetric (0, 2)-tensor field c on M is a holomorphic tensor field with respect to the almost complex structure J . P r o o f. Let (M,J, g) be a Kähler–Norden manifold. Put A ( X̃, Ỹ ) = g̃∇,c ( HJX̃, Ỹ ) − g̃∇,c ( X̃,H JỸ ) for any X̃, Ỹ ∈ =1 0 (T ∗M). For all vector fields X̃ and Ỹ , which are of the form V ω, V θ or HX, HY , from (3.2) and (3.3), we have A ( HX,H Y ) = g̃∇,c ( HJ(HX),H Y )− g̃∇,c ( HX,H J(HY ) ) = g̃∇,c ( H(JX),H Y )− g̃∇,c ( HX,H (JY ) ) = c(JX, Y )− c(X, JY ), A ( HX,V θ ) = g̃∇,c ( HJ(HX),V θ )− g̃∇,c ( HX,H J(V θ) ) = g̃∇,c ( H(JX),V θ )− g̃∇,c ( HX,V (θ ◦ J) ) = θ(JX)− (θ ◦ J)(X), A ( V ω,H Y ) = g̃∇,c ( HJ(V ω),H Y )− g̃∇,c ( V ω,H J(HY ) ) = (ω ◦ J)(Y )− ω(JY ), A ( V ω,V θ ) = g̃∇,c ( HJ(V ω),V θ )− g̃∇,c ( V ω,H J(V θ) ) = g̃∇,c ( HJ(V ω),V θ )− g̃∇,c ( V ω,H J(V θ) ) = g̃∇,c ( V (ω ◦ J),V θ )− g̃∇,c ( V ω,V (θ ◦ J) ) = 0. 164 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions From the last equations, if the symmetric (0, 2)−tensor field c is pure with respect to J , we say that A ( X̃, Ỹ ) = 0, i.e., the modified Riemannian extension g̃∇,c is pure with respect to HJ . Now we are interested in the holomorphy property of the modified Riemannian extension g∇,c with respect to HJ . We calculate (ΦHJ g̃∇,c)(X̃, Ỹ , Z̃) = (HJX̃)(g̃∇,c(Ỹ , Z̃))− X̃(g̃∇,c(HJỸ , Z̃)) + g̃∇,c((LỸ HJ)X̃, Z̃) + g̃∇,c(Ỹ , (LZ̃ HJ)X̃) for all X̃, Ỹ , Z̃ ∈ =1 0(T ∗M). Then we obtain the following equations: (ΦHJ g̃∇,c)(V ω, V θ, HZ) = 0, (ΦHJ g̃∇,c)(V ω, V θ, V σ) = 0, (ΦHJ g̃∇,c)(V ω, HY, V σ) = 0, (ΦHJ g̃∇,c)(V ω, HY, HZ) = (ω ◦ ∇Y J)(Z) + (ω ◦ ∇ZJ)(Y ), (ΦHJ g̃∇,c)(HX, V ω, HZ) = (ΦJg)(X, ω̃, Z)− g((∇ω̃J)X,Z), (ΦHJ g̃∇,c)(HX, V ω, V σ) = 0, (ΦHJ g̃∇,c)(HX, HY, HZ) = (ΦJc)(X,Y, Z)) +(p ◦R(Y, JX)− p ◦R(Y,X)J)(Z) +(p ◦R(Z, JX)− p ◦R(Z, X)J)(Y ), (ΦHJ g̃∇,c)(HX, HY, V σ) = (ΦJg)(X, Y, σ̃)− g(Y, (∇σ̃J)X), where ω̃ = g−1 ◦ω = gijωj is the associated vector field of ω. On the other hand, the Riemannian curvature R of Kähler–Norden manifolds is pure [10], that is, R(JX, Y ) = R(X, JY ) = R(X, Y )J = JR(X, Y ). Hence, from the equations above, it follows that ΦHJ g̃∇,c = 0 if and only if ΦJc = 0, which completes the proof. 4. Curvature Properties of the Levi–Civita Connection of the Modified Riemannian Extension g̃∇,c In this section, we give the conditions under which the cotangent bundle T ∗M equipped with the modified Riemannian extension g̃∇,c is respectively locally flat, locally symmetric, conformally flat, projectively flat, semi-symmetric and Ricci semi-symmetric. Let us consider T ∗M equipped with the modified Riemannian extension g̃∇,c for a given symmetric connection ∇ on M . By virtue of (2.5) and (2.6), the mod- ified Riemannian extension (g̃∇,c)βγ and its inverse (g∇,c)βγ have the following Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 165 A. Gezer, L. Bilen, and A. Cakmak components with respect to the adapted frame {Eα}: (g̃∇,c)βγ = ( cij δj i δi j 0 ) , (4.1) (g∇,c) βγ = ( 0 δi j δj i −cij ) . (4.2) Theorem 2. Let ∇ be a symmetric connection on M and T ∗M be the cotan- gent bundle with the modified Riemannian extension g̃∇,c over (M,∇). Then i) (T ∗M, g̃∇,c) is locally flat if and only if (M,∇) is locally flat and the com- ponents cij of c satisfy the condition ∇i(∇kcjh −∇hcjk)−∇j(∇kcih −∇hcik) = 0; (4.3) ii) (T ∗M, g̃∇,c) is locally symmetric if and only if (M,∇) is locally symmetric and the components cij of c satisfy the condition ∇l∇i(∇kcjh −∇hcjk)−∇l∇j(∇kcih −∇hcik) −R m ijk (∇lcmh)−R m ijh (∇lckm) = 0. (4.4) P r o o f. The Levi–Civita connection ∇̃ of g̃∇,c is characterized by the Koszul formula 2g̃∇,c(∇̃X̃ Ỹ , Z̃) = X̃(g̃∇,c(Ỹ , Z̃)) + Ỹ (g̃∇,c(Z̃, X̃))− Z̃(g̃∇,c(X̃, Ỹ )) −g̃∇,c(X̃, [Ỹ , Z̃]) + g̃∇,c(Ỹ , [Z̃, X̃]) + g̃∇,c(Z̃, [X̃, Ỹ ]) for all X̃, Ỹ and Z̃ ∈ =1 0 (T ∗M). Using (4.1), (4.2) and Lemma 1, the following formulas can be checked by a straightforward computation: ∇̃Ei Ej = 0, ∇̃Ei Ej = 0, ∇̃EiEj = −Γj ihEh, ∇̃EiEj = Γh ijEh + {psR s hji + 1 2 (∇icjh +∇jcih −∇hcij)}Eh, (4.5) where R s hji are the components of the curvature tensor field R of the symmetric connection ∇ on M . The Riemannian curvature tensor R̃ of T ∗M with the modified Riemannian extension g̃∇,c is obtained from the well-known formula R̃ ( X̃, Ỹ ) Z̃ = ∇̃ X̃ ∇̃ Ỹ Z̃ − ∇̃ Ỹ ∇̃ X̃ Z̃ − ∇̃[X̃,Ỹ ]Z̃ 166 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions for all X̃, Ỹ , Z̃ ∈ =1 0(T ∗M). Then from Lemma 1 and (4.5), after standard computations, the Riemannian curvature tensor R̃ is obtained as follows: R̃(Ei, Ej)Ek = R h ijk Eh (4.6) +{ps(∇iR s hkj −∇jR s hki ) + 1 2 {∇i(∇kcjh −∇hcjk)−∇j(∇kcih −∇hcik), −R m ijk cmh −R m ijh ckm}}Eh, R̃(Ei, Ej)Ek = R k jih Eh, R̃(Ei, Ej)Ek = −R j hki Eh, R̃(Ei, Ej)Ek = R i hkj Eh, R̃(Ei, Ej)Ek = 0, R̃(Ei, Ej)Ek = 0, R̃(Ei, Ej)Ek = 0, R̃(Ei, Ej)Ek = 0 with respect to the adapted frame {Eα}. i) We now assume that R = 0 and equation (4.3) holds, then from the equa- tions in (4.6) it follows that R̃ = 0. Conversely, under the assumption that R̃ = 0, we evaluate the first equation in (4.6) at an arbitrary point (xi, pi) = (xi, 0) in the zero section of T ∗M and we have 0 = [R̃(Ei, Ej)Ek](xi,0) = R h ijk Eh + {1 2 {∇i(∇kcjh −∇hcjk)−∇j(∇kcih −∇hcik) −R m ijk cmh −R m ijh ckm}}Eh from which we get R = 0 and ∇i(∇kcjh −∇hcjk)−∇j(∇kcih −∇hcik) = 0. ii) We consider the components of ∇̃R̃. Using (4.5) and (4.6), by a direct computation, we obtain the following relations: ∇̃lR̃ h ijk = ∇lR h ijk , ∇̃lR̃ h ijk = ps(∇l∇iR s hkj −∇l∇jR s hki ) + 1 2 {∇l∇i(∇kcjh −∇hcjk) −∇l∇j(∇kcih −∇hcik)− (∇lR m ijk )cmh −R m ijk (∇lcmh) −(∇lR m ijh )ckm −R m ijh (∇lckm)}, ∇̃lR̃ h ijk = ∇lR k jih , ∇̃lR̃ h ijk = −∇lR j hki , ∇̃lR̃ h ijk = ∇lR i hkj , ∇̃lR̃ h ijk = ∇iR l hkj −∇jR l hki , Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 167 A. Gezer, L. Bilen, and A. Cakmak all the others being zero with respect to the adapted frame {Eα}. With the same method as i), the proof follows from the above equations. We turn our attention to the Ricci tensor of the modified Riemannian exten- sion g̃∇,c. Let R̃αβ = R̃ σ σαβ denote the Ricci tensor of the modified Riemannian extension g̃∇,c. From (4.6), the components of the Ricci tensor Rαβ are charac- terized by R̃jk = Rjk + Rkj R̃jk = 0, R̃jk = 0, R̃jk = 0, (4.7) with respect to the adapted frame {Eα}. Theorem 3. Let ∇ be a symmetric connection on M and T ∗M be the cotan- gent bundle with the modified Riemannian extension g̃∇,c over (M,∇). Then (T ∗M, g̃∇,c) is Ricci flat if and only if the Ricci tensor of ∇ is skew symmetric (for the Riemannian extension, see [15]). P r o o f. The proof follows from (4.7). Theorem 4. Let ∇ be a symmetric connection on M and T ∗M be the cotan- gent bundle with the modified Riemannian extension g̃∇,c over (M,∇), then (T ∗M, g̃∇,c) is a space of constant scalar curvature 0. P r o o f. The scalar curvature of the modified Riemannian extension g̃∇,c is defined by r̃ = (g̃∇,c)αβ R̃αβ . Using (4.2) and (4.7), we get r̃ = (g̃∇,c)ijR̃ij + (g̃∇,c)ijR̃ij + (g̃∇,c)ijR̃ij + (g̃∇,c)ijR̃ij = 0. R e m a r k 1. Let ∇ be a symmetric connection on M and T ∗M be the cotangent bundle with the modified Riemannian extension g̃∇,c over (M,∇). The cotangent bundle T ∗M with the modified Riemannian extension g̃∇,c is locally conformally flat if and only if its Weyl tensor W̃ vanishes, where the Weyl tensor is given by W̃αβγσ = R̃αβγσ + r̃ 2(2n− 1)(n− 1) {(g̃∇,c)αγ(g̃∇,c)βσ − (g̃∇,c)ασ(g̃∇,c)βγ} − 1 2(n− 1) ((g̃∇,c)βσR̃αγ−(g̃∇,c)ασR̃βγ+(g̃∇,c)αγR̃βσ−(g̃∇,c)βγR̃ασ) 168 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions and R̃αβγσ = R̃ λ αβγ (g̃∇,c)λσ. In [2], it is proved that (T ∗M, g̃∇,c) is locally con- formally flat if and only if (M,∇) is projectively flat and the components cij of c satisfy the condition ∇i(∇kcjn −∇ncjk)−∇j(∇kcin −∇ncik)−R h ijk chn −R h ijn ckh = 0. (4.8) Theorem 5. Let ∇ be a symmetric connection on M and T ∗M be the cotan- gent bundle with the modified Riemannian extension g̃∇,c over (M,∇). Then (T ∗M, g̃∇,c) is projectively flat if and only if (M,∇) is flat and the components cij of c satisfy the condition ∇i(∇kcjn −∇ncjk)−∇j(∇kcin −∇ncik) = 0. (4.9) P r o o f. A manifold is said to be projectively flat if the projective curvature tensor vanishes. The projective curvature tensor is defined by P̃αβγσ = R̃αβγσ − 1 (2n− 1) ((g̃∇,c)ασR̃βγ − (g̃∇,c)βσR̃αγ), where R̃αβγσ = R̃ λ αβγ (g̃∇,c)λσ. The non-zero components of projective curvature tensor of the modified Rie- mannian extension g̃∇,c are given by P̃ijkn = R h ijk chn + ps(∇iR s nkj −∇jR s nki ) + 1 2 {∇i(∇kcjn −∇ncjk)−∇j(∇kcin −∇ncik)−R h ijk chn −R h ijn ckh} − 1 (2n− 1) (cin(Rjk + Rkj)− cjn(Rik + Rki)), P̃ijkn = R n ijk − 1 (2n− 1) ( δn i (Rjk + Rkj)− δn j (Rik + Rki), P̃ijkn = R k jin , P̃ijkn = R j kni + 1 (2n− 1) δj n(Rik + Rki), P̃ijkn = R i nkj − 1 (2n− 1) δi n(Rjk + Rkj). The proof follows from the above equations. Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 169 A. Gezer, L. Bilen, and A. Cakmak A semi-Riemannian manifold (M, g), n = dim(M) ≥ 3, is said to be semi- symmetric [18] if its curvature tensor R satisfies the condition (R(X, Y )R)(Z, W )U = 0, (4.10) and Ricci semi-symmetric if its Ricci tensor satisfies the condition (R(X,Y )Ric)(Z, W ) = 0 (4.11) for all X, Y, Z, W,U ∈ =1 0(M), where R(X, Y ) acts as a derivation on R and Ric. In local coordinate, conditions (4.10) and (4.11) are respectively written in the following form: ((R(X,Y )R)(Z, W )U) n ijklm = ∇i∇jR n klm −∇j∇iR n klm = R n ijp R p klm −R p ijk R n plm −R p ijl R n kpm −R p ijm R n klp and ((R(X, Y )Ric)(Z,W ))ijkl = ∇i∇jRkl −∇j∇iRkl = R p ijk Rpl + R p ijl Rkp. Note that a locally symmetric manifold is obviously semi-symmetric, but in gen- eral the converse is not true. Theorem 6. Let (M, g) be a semi-Riemannian manifold and T ∗M be the cotangent bundle with the modified Riemannian extension g̃∇,c over (M, g). We assume that R̃ h ijk = 0, from which it follows that ∇iR s hkj − ∇jR s hki = 0 and ∇i(∇kcjh − ∇hcjk) − ∇j(∇kcih − ∇hcik) − R m ijk cmh − R m ijh ckm = 0, where R and R̃ are the curvature tensors of the Levi–Civita connections ∇ and ∇̃ of g and g̃∇,c, respectively. Then (T ∗M, g̃∇,c) is semi-symmetric if and only if (M, g) is semi-symmetric. P r o o f. We consider the condition (R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ = 0 for all X̃, Ỹ , Z̃, W̃ , Ũ ∈ =1 0(T ∗M). The tensor (R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ has the components ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) ε αβγθσ = R̃ ε αβτ R̃ τ γθσ − R̃ τ αβγ R̃ ε τθσ − R̃ τ αβθ R̃ ε γτσ − R̃ τ αβσR̃ ε γθτ (4.12) with respect to the adapted frame {Eα}. For the case of α = i, β = j, γ = k, θ = l, σ = m, ε = n in (4.12), it follows that ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) n ijklm = R̃ n ijp R̃ p klm + R̃ n ijp R̃ p klm − R̃ p ijk R̃ n plm − R̃ p ijk R̃ n plm −R̃ p ijl R̃ n kpm − R̃ p ijl R̃ n kpm − R̃ p ijm R̃ n klp − R̃ p ijm R̃ n klp = −R m ijp R p kl n −R p ijk R m pl n −R p ijl R m kpn −R p ijn R m klp = −((R(X,Y )R)(Z, W )U) m ijkl n . (4.13) 170 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 Properties of Modified Riemannian Extensions For the case of α = i, β = j, γ = k, θ = l, σ = m, ε = n in (4.12), we get ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) n ijklm = R̃ n ijp R̃ p klm + R̃ n ijp R̃ p klm − R̃ p ijk R̃ n plm − R̃ p ijk R̃ n plm −R̃ p ijl R̃ n kpm − R̃ p ijl R̃ n kpm − R̃ p ijm R̃ n klp − R̃ p ijm R̃ n klp = −R k ijp R p nlm −R p ijn R k plm −R p ijl R k npm −R p ijm R k nlp = −((R(X, Y )R)(Z,W )U) k ijnlm . (4.14) For the case of α = i, β = j, γ = k, θ = l, σ = m, ε = n in (4.12), we have ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) n ijklm = 0. (4.15) For the case of α = i, β = j, γ = k, θ = l, σ = m, ε = n in (4.12), we obtain ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) n ijklm = R̃ n ijp R̃ p klm + R̃ n ijp R̃ p klm − R̃ p ijk R̃ n plm − R̃ p ijk R̃ n plm −R̃ p ijl R̃ n kpm − R̃ p ijl R̃ n kpm − R̃ p ijm R̃ n klp − R̃ p ijm R̃ n klp . (4.16) For the case of α = i, β = j, γ = k, θ = l, σ = m, ε = n in (4.12), we obtain ((R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ) n ijklm = R̃ n ijp R̃ p klm + R̃ n ijp R̃ p klm − R̃ p ijk R̃ n plm − R̃ p ijk R̃ n plm −R̃ p ijl R̃ n kpm − R̃ p ijl R̃ n kpm − R̃ p ijm R̃ n klp − R̃ p ijm R̃ n klp = R i npj R p klm −R i pkj R p nml + R i plj R p nmk −R i pmj R p lkn = ((R(X, Y )R)(Z,W )U) i nmlkj − ((R(X, Y )R)(Z,W )U) k kl nmj . (4.17) The other coefficients of (R̃(X̃, Ỹ )R̃)(Z̃, W̃ )Ũ reduce to one of (4.16), (4.14) or (4.15) by the property of the curvature tensor. The proof follows from (4.13)– (4.17). Theorem 6 immediately gives the following result. Corollary 1. Let (M, g) be a semi-Riemannian manifold and T ∗M be the cotangent bundle with the modified Riemannian extension g̃∇,c over (M, g). If (M, g) is locally symmetric and the components cij of c satisfy the condition ∇i(∇kcjh −∇hcjk)−∇j(∇kcih −∇hcik)−R m ijk cmh −R m ijh ckm = 0, (4.18) then (T ∗M, g̃∇,c) is semi-symmetric. Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 171 A. Gezer, L. Bilen, and A. Cakmak Theorem 7. Let (M, g) be a semi-Riemannian manifold and T ∗M be the cotangent bundle with the modified Riemannian extension g̃∇,c over (M, g). Then (T ∗M, g̃∇,c) is Ricci semi-symmetric if and only if (M, g) is Ricci semi-symmetric. P r o o f. We study the condition (R̃(X̃, Ỹ )R̃ic)(Z̃, W̃ ) = 0 for all X̃, Ỹ , Z̃, W̃ ∈ =1 0(T ∗M). The tensor (R̃(X̃, Ỹ )R̃ic)(Z̃, W̃ ) has the components ((R̃(X̃, Ỹ )R̃ic)(Z̃, W̃ ))αβγθ = R̃ ε αβγR̃εθ + R̃ ε αβθR̃γε. (4.19) By putting α = i, β = j, γ = k, θ = l in (4.19), it follows that ((R̃(X̃, Ỹ )R̃ic)(Z̃, W̃ ))ijkl = R̃ p ijk R̃pl + R̃ p ijl R̃kp = 2R p ijk Rpl + 2R p ijl Rkp = 2((R(X,Y )Ric)(Z, W ))ijkl, all the others being zero. This finishes the proof. R e m a r k 2. i) If cij = 0, then conditions (4.3), (4.4), (4.8), (4.9) and (4.18) are identically fulfilled. ii) If cij is parallel with respect to ∇, then conditions (4.3), (4.4), (4.8), (4.9) and (4.18) are identically fulfilled. iii) If cij satisfies the relation ∇icjk −∇jcik = ∇kωij , where the components ωij define a 2–form on M and if (M,∇) is flat, then conditions (4.3), (4.4), (4.8), (4.9) and (4.18) are identically verified. Acknowledgement. The authors would like to thank the anonymous re- viewer for his/her valuable comments and suggestions to improve the quality of the paper. References [1] S. Aslanci, S. Kazimova, and A.A. Salimov, Some Remarks Concerning Riemannian Extensions. — Ukrainian Math. J. 62 (2010), No. 5, 661–675. 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